Abstract. This paper describes a simple candidate one-way hash func-tion which satisfies a quasi-commutative property that allows it to be used aa an accumulator. This property allows protocols to be developed in which the need for a trusted central authority can be eliminated. Space-efficient distributed protocols are given for document time stamping and for membership testing, and many other applications are possible. 1

We investigate the behaviour of P (R r) and P (R r) as r !1 for the random variable R := P 1 n=1 Qn Q n 1 k=1 M k , where ((Q k ; M k )) k2N is an independent, identically distributed sequence with P ( 1 M 1) = 1. Random variables of this type appear in insurance mathematics, as solutions of stochastic difference equations, in the analysis of probabilistic algorithms and elsewhere. Exponential and Poissonian tail behaviour can arise.

An algorithm is developed for the exact simulation from distributions that are defined as fixed-points of maps between spaces of probability measures. The fixed-points of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixedpoints with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method. AMS subject classifications. Primary: 65C10; secondary: 65C05, 68U20, 11K45.

Abstract. In this paper, we give a simple algorithm for sampling from the Dickman distribution. It is based on coupling from the past with a suitable dominating Markov chain.

An algorithm is developed for the exact simulation from distributions that are defined as fixed-points of maps between spaces of probability measures. The fixed-points of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. The sampling algorithm relies on a modified rejection method.

Compounding processes, also known as perpetuities, play an important role in many applications; in particular, in time series analysis and mathematical finance. Apart from some special cases, the distribution of a perpetuity is hard to compute, and large deviations estimates sometimes involve complicated constants which depend on the complete distribution. Motivated by this, we propose provably efficient importance sampling algorithms which apply to qualitatively different cases, leading to light and heavy tails. Both algorithms have the non-standard feature of being statedependent. In addition, in order to verify the efficiency, we apply recently developed techniques based on Lyapunov inequalities. 1

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